Question: s involving definite integrals (algebraic) AP.CALC: CHA‑4 (EU), CHA‑4.D (LO), CHA‑4.D.1 (EK), CHA‑4.D.2 (EK), CHA‑4.E (LO), CHA‑4.E.1 (EK) Google Classroom Facebook Twitter Email You might need: Calculator Problem The greater the daily dose of vitamin C patients take, the greater their plasma concentration of vitamin C becomes. This relationship increases at a rate of $0.1e^{-0.002x}$ micromoles per milligram of daily dose. By approximately how many micromoles does the plasma concentration increase between $x=100$ and $x=200$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $0.067$ (Choice B) B $0.15$ (Choice C) C $7.42$ (Choice D) D $33.5$
Explanation: Letting $c(x)$ be the plasma concentration with a dose of $x$ milligrams, we are given that $c'(x)=0.1e^{-0.002x}$. We want to find $c(200)-c(100)$. According to the Fundamental Theorem of Calculus, $\begin{aligned} c(200)-c(100)&=\int_{100}^{200} c'\left(x\right)dx \\\\ &=\int_{100}^{200}\left(0.1e^{-0.002x}\right)dx \end{aligned}$ $\int_{100}^{200}\left(0.1e^{-0.002x}\right)dx\approx6.88$ In conclusion, between $x=100$ and $x=200$ the plasma concentration increases by approximately $7.42$ micromoles.